CHAPTER 3 Flashcards by Sonnie Boy Balictar

GAMBLER TOSS COIN

A gambler devised a game to be played with a friend. He bet 1/2 the money in his pocket on the toss of a coin; heads he won, tails he lost. The coin was tossed and the money handed over. The offer was repeated and the game continued. Each time the bet was for 1/2 the money then in his possession. Eventually the number of times he lost was equal to thenumber of times he won. Quickly now! Did he gain, lose, or break even?

He lost, even if they played only twice, or four times, or six,
or…

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PRISONER, WHITE AND BLACK BALLS

A prisoner is given 10 white balls, 10 black balls and two boxes. He is told that an executioner will draw one ball from one of the two boxes. If it is white, the prisoner will go free; if it is black, he will die. How should the prisoner arrange the balls in the boxes to give himself the best chance for survival?

balls in 1st box?
balls in 2nd box?
survival rate?

1st box - 1 white
2nd box - 9 white & 10 black
survival: 73.7%

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MARKSMEN SHOOT EACH OTHER

Smith and Jones, both 50% marksmen, decide to fight a duel in which they exchange alternate shots until one is hit. What are the odds in favor of the man who shoots first?

2/3

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CIRCLE AND LINE

There are n points on a circle. A straight line segment is drawn between each pair of points. How many intersections are there within the circle if no 3 lines are collinear?

equation?

nC4
n(n-1)(n-2)(n-3) / 4!
where: n = number of points

example:
4 points = 1 intersection
5 points = 5 intersetion

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PARKING LOT

On a certain day, our parking lot contains 999 cars, no two of which have the same 3 digit license number. After 5:00 pm, what is the probability that the license numbers of the first 4 cars to leave the parking lot are in increasing order of magnitude?

1/24

notes:
4! or 24 possible permutations of 4 cars
Only one of these is in increasing rank of licence magnitude

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MARKSMEN SHOOT SPHERICAL TARGET

Three marksmen simultaneously shoot at and hit a rapidly spinning spherical target. What is the probability that the three points of impact are on the same hemisphere?

probability = 1

note:
since any three points on the surface of a sphere are always located on some hemisphere

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HOSPITAL NURSERY

A hospital nursery contains only two baby boys; the girls have not yet been counted. At 2:00 pm, a new baby is added to the nursery. A baby is then selected at random to be the first to have its footprint taken. It turns out to be a boy. What is the probability that the Iast addition to the nursery was a girl?

2/5

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SINKING SUBMARINE

Assume that a single depth charge has a probability of 1/2 of sinking a submarine, 1/4 of damage and 1/4 of missing. Assume also that two damaging explosions sink the sub. What is the probability that 4 depth charges will sink the sub?

probability to sunk?
probability to escape?

Probability (sunk) = 251/256 = 98%
Probability (escape) = 5/256 = 2%

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SUITES IN HAND AT BRIDGE

What is the most likely distribution of the suits in a hand at Bridge?

4-4-3-2 distribution

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TRUE COIN

Using a “true” coin, a random sequence of binary digits can be generated by letting, say heads denote zero and tails, one. An operations analyst wished to obtain such a sequence, but he had only one coin which he suspected was not true. Could he still do it?

could he still do it?
consider what throw?
delete?
equiprobable?

Conditions
* consider throws in pair
* delete occurence HH or TT
* HT and TH equiprobable, one may be used to denote 0, the other 1

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MARBLES IN THE BAG

If 2 marbles are removed at random from a bag containing black and white marbles, the chance that they are both white is 1/3. If
3 are removed at random, the chance that they all are white is 1/6. How many marbles are there of each color?

white?
black?

white = 6
black = 4

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2 BROTHER’S IN LAW

Rigorously speaking, two men are “brothers in-law” if one is married to the full sister of the other. How many men can there be with each man a brother-in-law of every other man?

3 men only in a group is possible

An additional mutual brother-in-law, however, is not possible without violating either the laws of bigamy or consanguinity.

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6 TO 5 ODDS

An expert gives team A only a 40% chance to win the World Series. Basing his calculation on this, a gambler offers 6 to 5 odds on team B to win the first game. Is his judgment sound?

chance team A?
chance team B?
odds 6 to 5?

the gambler is on the safe side.

notes:
A chance of winning = 0.4539
B chance of winning = 0.5461
Odds 6 to 5 = 0.5455

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SALESMAN CIRCLING 10 CITIES

A salesman visits ten cities arranged in the form of a circle, spending a day in each. He proceeds clockwise from one city to the next, except whenever leaving the tenth city he may go to either the first or jump to the second city. How many days must elapse before his location is completely indeterminate, i.e., when he could be in any one of the ten cities?

83 days

On the 82nd day it could be definitely stated that he was not in the first city

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COIN UNBALANCED

A coin is so unbalanced that you are likely to get two heads in two successive throws as you are to get tails in one. What is the probability of getting heads in a single throw?

0.618

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TIC TAC TOE, DART PLAYERS

Three dart players threw simultaneously at a tic-tac-toe board, each hitting a different square. What is the probability that the three hits constituted a win at tic-tac-toe?

2/21 = 9.5%

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SWIMMING POOL BUILDERS

Four swimming pool builders submit sealed bids to a homeowner who is required by law to accept the last bid that he sees, i.e., once he looks at a bid, he automatically rejects all previous bids. He is not required to open all the envelopes, of course. Assuming that all four bids are different, what procedure will maximize his chances of accepting the lowest bid. and what will be the probability of doing so?

uses the first bid as standard?
uses the first two bids as standard?

uses the first bid as standard = 11/24 = 46%
uses the first two bids as standard = 10/24 = 42%

FRATERNITY, BASKETBALL, AND HOCKEY

All the members of a fraternity play basketball while all but one play ice hockey; yet the number of possible basketball teams (5 mebers) is the same as the number of possible ice hockey teams (6 members). Assuming there are enough members to form either type of team, how many are in the fraternity?

members?
how many teams?

15 members
can field 3003 teams of either type

SUPER DOMINO

A game of super-dominoes is played with pieces divided into three cells instead of the usual two, containing all combinations from triple blank to triple six, with no duplications. For example the set does not include both 1, 2, 3 and 3, 2, 1 since these are merely reversals of each other. (But, it does contain 1, 3, 2.) How many pieces are there in a set?

how many pieces?
how many possibilities?
how many read same backward and forward?
how many are eliminated since they are duplicated?

196 pieces

notes:
343 possibilities
49 read same backward and forward
294 eliminated since they are duplicated

3 SIDED MARTIAN COINS

Martian coins are 3-sided (heads, tails, and torsos), each side coming up with equal probability. Three Martians decided to go odd-man-out to determine who pays a dinner check. (If two coins come up the same and one different, the owner of the latter coin foots the bill). What is the expected number of throws needed in order to determine a loser?

throws determine a loser?
probability all three coins are same?
probability all three coins are different?

throws determine a loser = 1 1/2

notes:
probability all three coins are same = 1/9 = 11.11%
probability all three coins are different = 2/9 = 22.22%

THREE FAMILIES, 2 SONS, 2 DAUGHTERS EACH

There are three families, each with two sons and two daughters. In how many ways can all these young people be married?

3 DIGIT TELEPHONE AREA CODES

How many three digit telephone area codes are possible given that:
(a) the first digit must not be zeto or one;
(b) the second digit must be zero or one;
(c) the third digit must not be zero;
(d) the third digit may be one only if the second digit is zero

136 possible codes

SIX MEN, SIX GUN, RUSSIAN ROULETTE

Six men decide to play Russian roulette with a six gun loaded with one cartridge. They draw for position, and afterwards’ the sixth man casually suggests that instead of letting the chamber rotate in sequence, each man spin the chamber before shooting. How would this improve his chances?

survival probability enhanced by?
probability firing fatal bullet?
probability first 5 men survive?
probability firing being shot after 5 men?

survival probability enhanced by 0.1 or 10% by spinning

notes:
probability firing fatal bullet = 1/6 = 16.67%
probability first 5 men survive = (5/6)^5 = 40%
probability firing being shot after 5 men = (5/6)^5 * 1/6 = 6.7

LONG SHOT POKER, PAT HAND

A long shot poker player draws two cards to the five and six of diamonds and the joker. What are his chances of coming up with a pat hand?

probability of pat hand = 0.168 = 16.8 %

# **GAME OF CRAPS, 16 HARD WAY** In Puevigi, the game of craps is played with a referee calling the point by adding together the six faces (three on each die) visible from his vantage point. What is the probability of making 16 the hard way? (That is, by throwing two eights.)

probability of making 16 the hard way = 0

# **BERMUDA OR YELLOWSTONE'S OLD FAITHFUL** Max and his wife Min each toss a pair of dice to determine where they will spend their vacation. If either of Min's dice displays the same number of spots as either of Max's, she wins and they go to Bermuda. Otherwise, they go to Yellowstone. What is the chance they'll see "Old Faithful" this year? ## Footnote Yellowstone (Old Faithful)? Bermuda?

Yellowstone (Old Faithful) = 0.486 = 48.6% Bermuda = 0.514 = 51.4%

# **BOOKWORM EATS BOOK PAGES** There are four volumes of an encyclopedia on a shelf, each volume containing 300 pages, (that is, numbered 1 to 600), but these have been placed on the shelf in random order. A bookworm starts at the first page of Vol. 1 and eats his way through to the last page of Vol. 4. What is the expected number of pages (excluding covers) he has eaten through?

500 pages

# **3 SEXES VENUSIAN BAT FISH** Venusian batfish come in three sexes, which are indistinguishable (except by Venusian batfish). How many live specimens must our astronauts bring home in order for the odds to favor the presence of a "mated triple" with its promise of more little batfish to come? ## Footnote no of specimens? odds of mated triple?

4 specimens: odds of mated triple = 4/9 = 44.44% 5 specimens: odds of mated triple = 50/81 = 62%

# **12 WHITE BALLS, 3 BLACK BALLS, OPAQUE BOTTLE** In a carnival game, 12 white balls and 3 black balls are put in an opaque bottle, shaken up, and drawn out one at a time. The player gets 25 cents for each white ball which emerges before the first black ball. If he pays one dollar to play, how much can be he expect to win (or lose) on each game?

* average loss = a quarter or 25 cents notes: * In the long run, 3 black balls will occur equally spaced in the stream of balls. * Therefore, can expect three white balls to appear before the first black ball.

# **BASE TEN?** In the binary system there are only two positive integers containing no digit more than once, namely 1 and 10. How many are there in base ten?

8,677,690

# **MORE UNLIKELY BRIDGE?** Which is the more unlikely event in bridge: the ultimate in distribution (a 13 card suit) or the ultimate in point count?

* two events are equally improbable (NOT HAPPENING)

# **LONGEST LEGAL BRIDGE?** What is the longest legal sequence of bids in bridge hand? ## Footnote initial 3 passes + terminal pass? final 3 passes following the redouble of seven no-trump are academic?

Case 1: initial 3 passes + terminal pass * maximum total = **319** Case 2: final 3 passes following the redouble of seven no-trump are academic * actual maximum total = **316**

# **NBA WIN PERCENTAGE?** In the final seconds of the game, your favorite N.B,A. team is behind 117 to 118. Your center attempts a shot and is fouled for the 2nd time in the last 2 minutes as the buzzer sounds. Three to make two in the penalty situation. Optimistic? Note: the cenler is only a 50% free-thrower. What are your team's overall chances of winning?

11/16 or about 69%

# **THROWING 7 WITH 2 DICE?** One of a pair of dice is loaded so that the chance of a 1 turning up is 1/5, the other faces being equally likely. Its mate is loaded so that the chance of a 6 turning up is 1/5, the other faces equally likely. How much does this loading increase probability of throwing a 7 with the two dice?

1/750 = 0.13%

# **417TH TERM?** If all 720 permutations of the digits 1 through 6 are arranged in numerical order, what is the 417th term?

432516

# **FORECASTER VS. FEDERAL METEOROLOGICAL SERVICE** The local weather forecaster says "no rain" and his record is 2/3 accuracy of prediction. But the Federal Meteorological Service predicts rain and their record is 3/4. With no other data available, what is the chance of rain?

3/5

# **SHARP OPERATOR, TOSS COIN** A sharp operator makes the following deal. A player is to toss a coin and receive 1, 4, 9, ... n^2 dollars if the first head comes up on the first, second, third, . . . n-th toss. The sucker pays ten dollars for this. How much can the operator expect to make if this is repeated a great many times?

4 dollars per game on the average

# **SERIES ENDED WHERE IT BEGAN?** In 1969, the World Series began in the stadium of the American League pennant winner. Assume the contenders are evenly matched. What is the probability that the series ended where it began?

5/8 = 62.5%

# **CARNIVAL OPERATOR, BALLS** In a carnival game 5 balls are tossed into a square box divided into 4 square cells, with baffes to insure that every ball has an equal chance of going in any cell. The player pays $1 and receives $1 for every cell which is empty after the 5 balts are thrown. How much does the operator expect to make per game?

A nickel (5 cents) a game on the average

# **1 RED CHECKER THROWN OUT THE WINDOW?** Having lost a checker game, a specialist in learning programs threw one of the red checkers out the window. His wife reboxed the 12 black pieces and 11 red pieces one at a time in random fashion. The number of black checkers in the box always exceeded the number of reds. What was the a priority probability of this occurrence?

1/23